130 ANALYTICAL MECHANICS
3. Discuss the motions of a particle which is repelled from a fixed point with a force which varies directly as the distance of the particle from the fixed point.
112. III. Motion of a Particle in a Resisting Medium. — As a concrete example of motion in a resisting medium consider the motion of falling bodies, taking the resistance of the atmosphere into account. At any instant of the motion two forces act on the body, i.e., the weight of the body and the resistance of the air. Denoting the resisting force by F we get
dv T, /I,
m — = mg — F (1)
for the force equation. In order to be able to integrate the last equation we must make an assumption as to the nature of F.
CASE I. RESISTANCE PROPORTIONAL TO THE VELOCITY.. — Suppose F to be proportional to the velocity, then
F = kiVj
where ki is a positive constant. Substituting in the force equation this expression for F we obtain
dv^ _ dt
where k = —. Rearranging the last equation
dv __ __, ,
V~k Integrating
or